Question

The difference between two roots of equation 4x* - 4x - 25x + x + 6 = 0 is unity, then the
difference between other two roots is
1) 2
2) 3
3) 4
4) 5​

Answer 1

Answer:

4

Step-by-step explanation:

very simple ,

plz mark brainliest

Answer 2

Concept:

Examine the highest-degree term of the polynomial – that is, the term with the highest exponent. That exponent is how many roots the polynomial will have.

The roots of a polynomial are also called its zeroes, because the roots are the ​x​ values at which the function equals zero.

$a^{2}- b^{2}=(a+b)(a-b)$

Given:

A polynomial $4x^{4}-4x^{3}-25x^{2}+x+6$ and the diffeence betwwen 2 of it roots is given as 1.

To find:

We need to find the difference of the other two roots.

Solution:

Now lets assume the roots are a, b, c and d.

Lets solve the quadratic equation and find its roots.

x-3 )   $4x^{4}-4x^{3}-25x^{2}+x+6$   (  $4x^{3} +8x^{2} -x-2$

         $4x^{4}-12x^{3}$

      - ___________________             [subtracting]

                   $8x^{3}-25x^{2} +x+6$

                   $8x^{3}-24x^{2}$

     -  ___________________             [subtracting]

                             $-x^{2} +x+6$

                             $-x^{2} +3x$

     - ___________________             [subtracting]

                                      -2x+6

                                      -2x+6

                                - ______              [subtracting]

                                              0        

Therefore, the poynomial can be written as $(4x^{3} +8x^{2} -x-2)(x-3)$

Long division for $4x^{3} +8x^{2} -x-2$

x + 2  )  $4x^{3} +8x^{2} -x-2$  (  $4x^{2} -1$

            $4x^{3} +8x^{2}$

         - ______________               [subtracting]

                             - x - 2

                             - x - 2

        - ______________                [subtracting]            

                                    0

Therefore, the poynomial can be written as $(x-3)(4x^{2}-1)(x+2)$

Now,

$4x^{2}-1=(2x)^{2} - 1^{2}$

Therefore, the roots to $4x^{2} -1$ are (2x-1)(2x+1)

The equation can be written as (x-3)(x+2)(2x-1)(2x+1)

The roots are $3, -2, -\frac{1}{2} ,\frac{1}{2}$

Now on comparison we can say that

$\frac{1}{2} -(-\frac{1}{2})=1$

Hence the other 2 roots are 3, -2

The difference of theses roots are,

3 - (-2)=3+2=5

Therefore the difference of the other two roots is 5.

Correct option is (4)